"Quantum Topology" here means that the
quantum of action h generates a functional "quantum space" and
that the energy-momentum and space-time are dual coordinates that
live and get projected from that topological space which
represents the invariant arena in which physical interactions take
place. In quantum theory we have an abstract mathematical image of
that quantum manifold in the form of an antilinear-bilinear form;
the complete Dirac bracket,

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Quantum space provides a consistent mathematical scheme to
incorporate both theory of relativity represented by a space-time
manifold and quantum mechanics represented by a quantum dynamical
variable that is non-commutative with the manifold. The manifold
and quantum dynamics are connected in a mathematical manner
similar to the way vectors and their dual vectors are connected in
the theory of functional spaces. The relativistic manifold is
extended into a quantum manifold that incorporates quantum
dynamics, and commutation relations define topological structures
in the quantum manifold.

"Quantum Topodynamics"
incorporates the gauge interactions into the structure of the
quantum manifold through introducing a proper topological group
structure on the fundamental set of the quantum space. A quantum
set is defined as the 2-fold infinite set of the dual coordinates
of the quantum space D and Q provided by the Fourier
representation. Then, we study the continuous mathematical
transformations on the set that generate a topological group with
a compact graded Lie manifold and gauge field (fibre bundle
structure of the quantum space).

We represent the
continuous mapping on the set as a logical operation to represent
the algebraic structure as an orthomodular structure. This
approach maps the structure of the group into properties of the
logic. This gives us an insight into quantum computation and a
criterion for the finiteness of functional integration on the
basis of the global properties of the functional space. An
immediate application to the central role played by the fibre
bundle structure in quantum computation is to attempt to construct
a quantum processor along the sructure of the fibre bundle. By
representing the fundamental operation as quantum interference and
reflecting the group structure in a matrix quantum interference
devise, this matrix processor will allow NxN gauge potentials to
act on the phases of N rays and the interference of these rays
will generate a continuous holographic output that represents the
topology of the quantum state being computed from the functional
integral of quantum topodynamics.